3.4.2 1 sample variance

1 sample variance Purpose of the 1 sample variance test The 1 sample variance test is used to determine whether the variability of a single population is equal to a specified target (or known) variance. - Question answered: “Is the process variance equal to a specific value?” - Typical use: Compare current process variability to: - A design requirement - A historical variance - A customer specification for spread This test is based on the chi-square distribution and is applied to continuous, normally distributed data. --- Core concepts: variance and standard deviation Variance and standard deviation definitions Variance and standard deviation describe data spread around the mean. - Variance (σ² or s²): - Population variance: average squared distance of each value from the population mean - Sample variance: estimate of population variance from sample data - Standard deviation (σ or s): - Square root of variance - Expressed in the same units as the data The 1 sample variance test is directly performed on variance, but in practice you often start from the sample standard deviation. Population vs. sample variance - Population variance (σ²): - Parameter of the entire process or population - Unknown in most real settings - Sample variance (s²): - Calculated from a finite sample - Used as an estimator of σ² The 1 sample variance test uses s² from your sample to make an inference about σ². --- Assumptions of the 1 sample variance test Correct interpretation of the test requires specific conditions. - Normality: - The underlying population must be normally distributed - The chi-square approach is sensitive to non-normality - Independence: - Sample observations are independent of each other - Single population: - Data come from one consistent process or population When these assumptions are significantly violated, the chi-square test for variance may give misleading results. --- Hypotheses for the 1 sample variance test Null and alternative hypotheses You compare the unknown population variance σ² to a hypothesized variance σ₀². - Null hypothesis (H₀): σ² = σ₀² (The true variance equals the target or historical variance.) - Alternative hypothesis (H₁) can take three forms: - Two-sided: σ² ≠ σ₀² - Upper-sided: σ² > σ₀² - Lower-sided: σ² < σ₀² Choice of one-sided vs. two-sided depends on the practical question. Typical practical formulations - Concerned about increased variability: - H₀: σ² = σ₀² - H₁: σ² > σ₀² - Concerned about reduced variability (e.g., risk of over-tight control): - H₀: σ² = σ₀² - H₁: σ² < σ₀² - Concerned about any change in variability: - H₀: σ² = σ₀² - H₁: σ² ≠ σ₀² --- Test statistic and chi-square distribution Test statistic formula Given a sample of size n with variance s², and a hypothesized population variance σ₀², the test statistic is: - Chi-square statistic: χ² = (n − 1)·s² / σ₀² where: - n = sample size - s² = sample variance - σ₀² = hypothesized variance under H₀ Under the null hypothesis and normality, χ² follows a chi-square distribution with (n − 1) degrees of freedom. Degrees of freedom - Degrees of freedom (df): n − 1 - The chi-square distribution shape depends strongly on df: - For small df, it is skewed to the right - As df increases, it becomes more symmetric Correct df is essential for accurate critical values and p-values. --- Critical regions and decision rules Two-sided test For a significance level α: - Critical values: - Lower critical value: χ²(L) = χ²(α/2, df) - Upper critical value: χ²(U) = χ²(1 − α/2, df) - Decision rule (two-sided): - Reject H₀ if χ² ≤ χ²(L) or χ² ≥ χ²(U) - Fail to reject H₀ otherwise This checks for variance either significantly smaller or larger than σ₀². One-sided upper test For H₁: σ² > σ₀²: - Critical value: - Upper critical: χ²(U) = χ²(1 − α, df) - Decision rule: - Reject H₀ if χ² ≥ χ²(U) - Fail to reject H₀ otherwise This focuses only on detecting an increase in variance. One-sided lower test For H₁: σ² < σ₀²: - Critical value: - Lower critical: χ²(L) = χ²(α, df) - Decision rule: - Reject H₀ if χ² ≤ χ²(L) - Fail to reject H₀ otherwise This focuses only on detecting a decrease in variance. --- p-values for the 1 sample variance test Two-sided p-value For observed test statistic χ²(obs) and df: - If χ²(obs) > expected center (df): - p-value = 2 × [1 − F(χ²(obs))] - If χ²(obs) < expected center: - p-value = 2 × F(χ²(obs)) where F is the cumulative distribution function of the chi-square distribution with df degrees of freedom. One-sided p-values - Upper-sided (H₁: σ² > σ₀²): - p-value = P(χ² ≥ χ²(obs)) = 1 − F(χ²(obs)) - Lower-sided (H₁: σ² < σ₀²): - p-value = P(χ² ≤ χ²(obs)) = F(χ²(obs)) Interpretation: - Small p-value (≤ α): evidence against H₀ in favor of H₁ - Large p-value (> α): insufficient evidence to reject H₀ --- Confidence interval for the population variance Interval for variance A (1 − α)·100% confidence interval for σ² is: - Lower limit: (n − 1)·s² / χ²(1 − α/2, df) - Upper limit: (n − 1)·s² / χ²(α/2, df) where χ²(p, df) is the pth quantile of the chi-square distribution. Interval for standard deviation If needed, take the square root of each limit: - Lower limit for σ: √(lower variance limit) - Upper limit for σ: √(upper variance limit) The confidence interval provides a plausible range for the true variance (or standard deviation), not just a yes/no test outcome. --- Practical applications and interpretation When to use the 1 sample variance test Use this test when: - You have a single sample from a process - A target or historical variance (or standard deviation) is specified - You want to assess whether current variability matches that value Examples include checks on: - Process spread relative to a known historical level - Equipment performance compared with a specified variance - Measurement system variability relative to an acceptable limit (when modeled as a single variance comparison) Interpreting results for improvement - Reject H₀, σ² > σ₀²: - Evidence that the process is more variable than intended - Suggests need to: - Investigate special causes - Reduce sources of variation - Reject H₀, σ² < σ₀²: - Evidence that the process is less variable than the hypothesized value - Could support claims of improved consistency - Fail to reject H₀: - No strong statistical evidence of a variance difference from σ₀² - Does not prove equality, but suggests data are consistent with σ² = σ₀² In all cases, interpretation should consider: - Sample size and power - Assumption of normality - Practical significance, not only statistical significance --- Common pitfalls and checks Normality concerns The chi-square based 1 sample variance test assumes normality. - Use preliminary checks: - Histograms or probability plots - Knowledge of process behavior - If data are strongly non-normal: - Results of the chi-square variance test may be unreliable - Transformations or alternative methods may be necessary, but those go beyond this specific test Misuse of specification limits - The 1 sample variance test compares to a specific variance, not directly to specification limits. - Confusion to avoid: - Comparing variance to tolerance width without proper translation - If needed: - Convert specification-based expectations into a target variance or standard deviation and test against that value. --- Summary The 1 sample variance test assesses whether the variance of a single population matches a specified value using the chi-square distribution. It relies on a sample variance s², compares it to a hypothesized variance σ₀² through the statistic χ² = (n − 1)·s² / σ₀² with n − 1 degrees of freedom, and evaluates this against critical values or p-values under normality and independence assumptions. One- or two-sided hypotheses allow focus on increased, decreased, or any change in variability. Confidence intervals for the variance (and standard deviation) complement the test by providing a plausible range for the true variability. Proper application and interpretation require attention to assumptions, sample size, and the practical implications of differences in variance.